Sullivan Algebra and Trigonometry: Section 2.3 Lines Objectives Calculate and Interpret the Slope of a Line Graph Lines Given a Point and the Slope Use the Point-Slope Form of a Line Find the Equation of a Line Given Two Points Write the Equation of a Line in Slope-Intercept From and in General Form. Identify the Slope and the y Intercept of a Line from its Equation.

Let and be two distinct points with. The slope m of the non-vertical line L containing P and Q is defined by the formula If, L is a vertical line and the slope m of L is undefined (since this results in division by 0).

y x Slope can be though of as the ratio of the vertical change ( ) to the horizontal change ( ), often termed “rise over run”.

L y x If, then is zero and the slope is undefined. Plotting the two points results in the graph of a vertical line with the equation.

Example: Find the slope of the line joining the points (3,8) and (-1,2).

Some Important Facts about slope: 1. When the slope of a line is positive, the line slants upward from left to right. (L 1 ) 2. When the slope of a line is negative, the line slants downward from left to right. (L 2 ) 3. When the slope is zero, the line is horizontal. (L 3 ) 4. When the slope is undefined, the line is vertical. (L 4 ) L1L1 L2L2 L3L3 L4L4

Example: Draw the graph of the line passing through (1,4) with a slope of -3/2. Step 1: Plot the given point. Step 2: Use the slope to find another point on the line (vertical change = -3, horizontal change = 2). y x ( 1,4 ) 2 -3 (3,1)

Example: Draw the graph of the equation x = 2. y x x = 2

Theorem: Point-Slope Form of an Equation of a Line An equation of a non-vertical line of slope m that passes through the point (x 1, y 1 ) is:

Example: Find an equation of a line with slope -2 passing through (-1,5).

A horizontal line is given by an equation of the form y = b, where (0,b) is the y- intercept. Example: Graph the line y=4. y x y = 4

The equation of a line L is in general form with it is written as where A, B, and C are three real numbers and A and B are not both 0. The equation of a line L is in slope-intercept form with it is written as y = mx + b where m is the slope of the line and (0,b) is the y-intercept.

Example: Find the slope m and y-intercept (0,b) of the graph of the line 3x - 2y + 6 = 0.